Uniqueness of Simultaneity

Abstract

I invesigate the question of existence and uniqueness of simultaneity structures in spacetimes whose automorphism group, Aut, is either the inhomogeneous proper orthochronous Galilei or Lorentz group. An absolute simultaneity structure is defined as Aut-invariant equivalence relation whose equivalence classes are acausal sets. It is unique for Galilean and non-existent for Lorentzian spacetimes. Simultaneity relative to some additional structure X on spacetime is defined analogously, where Aut is now replaced with the stabilizer subgroup of X in Aut. It turns out that Einsteinian simultaneity is unique if X is an inertial frame (foliation by timelike straight lines). Finally I discuss the relation to work of others.